The number of possible outcomes is derived from tabulating all possible outcomes from a 3-dice roll. The total number of possible outcomes is 216 or 6 x 6 x 6 (this is the formula for a permutation where repetition is allowed).

Probability (Win) refers to the possible outcomes for each number divided by the total possible outcomes from a 3-dice roll.

Example:

For point 5, 6/216 = 0.0278

For point 10, 27/216 = 0.1250

Probability (Loss) is 1-Probability (Win).

Example:

For point 5, 1-0.0278=0.9722

For point 9, 1-0.1157=0.8843

Expectation is Probability (Win) – Probability (Loss).

Example:

For point 9, 0.1157 – 0.8843 = -0.7685

In the table the expectation has a ‘-‘ in front as this is in the perspective of the player having a negative expectation; which is positive for the house.

Break-even odds refers to the payment odds at which expectation is 0. This means that the house should ALWAYS pay BELOW these odds in order to maintain a positive expectation for the house.

The is derived from the Probability (Loss)/ Probability (Win),

Example:

For point 3, 0.9954/0.0046 = 215

For point 6, 0.9537/0.0463 = 20.6

After computing all the possible outcomes of a 3-dice roll, we can see that the possible outcomes and probability follow the central limit theorem nicely. The expectation dovetails the probability, being the result of 1-probability.

Single Die

For Single Die wagers, the math is different as a single wager wins when the specific number appears 1, 2 or 3 times. So, what we have to do is consider all the probabilities together.

Wager

Permutations

Probability

Break Even

Exp.

1 Die

75

0.3472

0.56

0.19292

2 Dice

15

0.0694

2.78

0.19292

3 Dice

1

0.0046

41.67

0.19292

Loss

125

0.5788

-1

-0.5788

Total Exp.

0

The method of getting the break even is the most straight-forward.

Divide the Probability of a Loss by the total number of winning wager types as below:

0.5788 / 3 = 0.19292

Then divide this by the probability of the individual winning wager:

Wager

Working

Probability

Break Even

1 Die

0.19292 / 0.3472

0.3472

0.5556551

2 Dice

0.19292 / 0.0694

0.0694

2.7798769

3 Dice

0.19292 / 0.00462963

0.00462963

41.671467

You will find that by multiplying each wagers probability by the break even, you would have the same number, except positive, as that of the probability of losing. Adding the 2 numbers together would give you an expectation of 0.

This is similar to slot mathematics, which we might get into later at some point.

Number of possible outcomes for each point value from 3 to 18

Probability for each point value from 3 to 18

Expectation for each point value from 3 to 18

The following table shows a similar tabulation for 3-Single Dice combinations and Double and Single Dice combinations. Again, the total possible number of outcomes is 216, with the number of categories being 56:

S/No

Combination

Possible Outcomes

Probability (Win)

Probability (Loss)

Expectation

Break-even odds

1

666

1

0.0046

0.9954

-0.9907

215

2

333

1

0.0046

0.9954

-0.9907

215

3

111

1

0.0046

0.9954

-0.9907

215

4

444

1

0.0046

0.9954

-0.9907

215

5

222

1

0.0046

0.9954

-0.9907

215

6

555

1

0.0046

0.9954

-0.9907

215

7

244

3

0.0139

0.9861

-0.9722

71

8

334

3

0.0139

0.9861

-0.9722

71

9

144

3

0.0139

0.9861

-0.9722

71

10

335

3

0.0139

0.9861

-0.9722

71

11

166

3

0.0139

0.9861

-0.9722

71

12

336

3

0.0139

0.9861

-0.9722

71

13

223

3

0.0139

0.9861

-0.9722

71

14

344

3

0.0139

0.9861

-0.9722

71

15

225

3

0.0139

0.9861

-0.9722

71

16

355

3

0.0139

0.9861

-0.9722

71

17

233

3

0.0139

0.9861

-0.9722

71

18

366

3

0.0139

0.9861

-0.9722

71

19

114

3

0.0139

0.9861

-0.9722

71

20

116

3

0.0139

0.9861

-0.9722

71

21

266

3

0.0139

0.9861

-0.9722

71

22

445

3

0.0139

0.9861

-0.9722

71

23

155

3

0.0139

0.9861

-0.9722

71

24

446

3

0.0139

0.9861

-0.9722

71

25

224

3

0.0139

0.9861

-0.9722

71

26

455

3

0.0139

0.9861

-0.9722

71

27

113

3

0.0139

0.9861

-0.9722

71

28

466

3

0.0139

0.9861

-0.9722

71

29

115

3

0.0139

0.9861

-0.9722

71

30

122

3

0.0139

0.9861

-0.9722

71

31

226

3

0.0139

0.9861

-0.9722

71

32

556

3

0.0139

0.9861

-0.9722

71

33

112

3

0.0139

0.9861

-0.9722

71

34

566

3

0.0139

0.9861

-0.9722

71

35

255

3

0.0139

0.9861

-0.9722

71

36

133

3

0.0139

0.9861

-0.9722

71

37

235

6

0.0278

0.9722

-0.9444

35

38

236

6

0.0278

0.9722

-0.9444

35

39

135

6

0.0278

0.9722

-0.9444

35

40

125

6

0.0278

0.9722

-0.9444

35

41

124

6

0.0278

0.9722

-0.9444

35

42

245

6

0.0278

0.9722

-0.9444

35

43

346

6

0.0278

0.9722

-0.9444

35

44

456

6

0.0278

0.9722

-0.9444

35

45

356

6

0.0278

0.9722

-0.9444

35

46

246

6

0.0278

0.9722

-0.9444

35

47

146

6

0.0278

0.9722

-0.9444

35

48

156

6

0.0278

0.9722

-0.9444

35

49

345

6

0.0278

0.9722

-0.9444

35

50

256

6

0.0278

0.9722

-0.9444

35

51

234

6

0.0278

0.9722

-0.9444

35

52

126

6

0.0278

0.9722

-0.9444

35

53

145

6

0.0278

0.9722

-0.9444

35

54

123

6

0.0278

0.9722

-0.9444

35

55

136

6

0.0278

0.9722

-0.9444

35

56

134

6

0.0278

0.9722

-0.9444

35

So, how do you calculate the combinations and permutations mathematically?

For combinations, we use 6 factorial or 6! as our numerator (due to the die being 6-sided). Our denominator would be a multiplication of the numbers involved in the combination and those not, with the total being 6 as well. Here’re some examples:

Remember that there are 6 numbers, so we have 6/6 = 1 combination for each number. This would apply in the next example too.

Specific Doubles: 6! / (1! X 1! X 4!) = 720 / (1 x 1 x 24) = 30 combinations (Notice how the numerator and denominator also equal 6? Notice the denominator 1! X 1! X 4!, as 1 of the numbers would be a double, 1 of the numbers would be a single and the other 4 numbers would not be in the combination.)

Remember that for Specific Doubles there are 6 numbers, so each number would have 30 / 6 = 5 combinations for each number.

3 Single Die Combination: 6!/(3! X 3!) = 720 / (6 x 6) = 720 / 36 = 20 combinations. (We have 3! as 3 of the numbers of the 6 are singles included in the combination, with the other 3 numbers out of the combination.)